An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Solution 1:

Not a proper answer, but it is the closest one to your integral that I was able to find. Using formula 7.141.1 in Gradshteyn-Ryzhik, after some simplifications, one can get: $$\int_0^\infty P_s(x+1)\,e^{-x}\,dx=\frac{e\,\sqrt2}{\sqrt\pi} K_{s+\frac{1}{2}}(1),$$ where $K_\nu(x)$ is the modified Bessel function of the 2nd kind. Note that this formula contains $(x+1)$ rather than $(x-1)$ that appears in your question. I'm still trying to find a proper answer...

Solution 2:

For $s=0,1,2,\ldots$ we compute $$ \mathcal{J}(s) = 1, 0, 1, 5, 36, 329, 3655, 47844, 721315, 12310199, 234615096, \ldots $$ [This was obtained with the gp code

F(p) = sum(k=0, poldegree(p), k!*polcoeff(p,k))
N = 10
vector(N+1,s,F(pollegendre(s-1,x-1)))

and can also be obtained from the expansion of $P_s(x-1)$ in powers of $x$, giving $(-1)^s \sum_{k=0}^s 2^{-k} k! {s \choose k} {-s-1 \choose k}$.] This matches OEIS sequence 806 up to sign, indicating that we're dealing with $(-1)^s y_s(-1)$ where $y_s$ is the Bessel polynomial of degree $s$. There's a lot of information about these numbers in that OEIS entry, including a recurrence equivalent to $$ {\mathcal J}(s) = (2s-1) {\mathcal J}(s-1) + {\mathcal J}(s-2) $$ and a simpler formula in terms of modified Bessel functions:

a(n) = BesselK[n+1/2,-1] / BesselK[5/2,-1]

contributed by Vaclav Kotesovec on Aug 07 2013. These formulas, together with known recurrences for modified Bessel functions, should soon also yield Vladimir Reshetnikov's conjectural formula in terms of $I_{s+1/2}(-1)$ and $K_{s+1/2}(1)$.